How can I create a library of animations?
Blender allows you to link or append.

Linking only creates a reference to the cross-referenced object.

Appending imports the cross-referenced object into your current file as a local copy.

Both are available through the File menu.

There's a great answer already on this site:

How can I reuse existing materials, objects, etc from an existing .blend?

Presets

So in response to your updated question I understand that you want something accessible from all projects without linking.

This is where add-ons come in.  There are loads of them out there and if you know (or are willing to learn) Python then you can roll your own add-on.  There are plenty of examples out there and here on how to go about this and also the source code for the included add-ons are available on git.blender.org.